Monotonicity Tests for Signal Deciles

The standard way to eyeball a signal is to sort stocks into ten deciles by signal value each month, then look at the average return of decile 10 minus decile 1. A big, positive gap looks like evidence the signal works. But that single comparison can hide a lot: what if the returns actually rise smoothly across deciles 1 through 7, then jump erratically, with decile 8 lower than decile 6? A large top-minus-bottom spread can be produced by a genuinely monotonic, well-behaved signal, or by a signal whose only real information is "the very top and very bottom deciles are different," with nothing systematic happening in between, two very different claims about how the signal works.

What monotonicity actually checks

A monotonicity test asks whether decile-average returns increase (or decrease) in the expected direction at every single step, not just between the extremes: is decile 2's average return higher than decile 1's, is decile 3's higher than decile 2's, and so on. This is often tested with a bootstrap-based test on the ordered sequence of decile means, checking whether a fully increasing sequence is unlikely to arise by chance, or with simpler nonparametric rank-based approaches.

A fully monotonic pattern says the signal captures something that scales gradually across its whole range, the behavior expected of a genuine, continuous relationship (cheaper stocks earning steadily higher returns as valuation decreases). A non-monotonic pattern, where the middle deciles are jumbled despite decile 10 beating decile 1, suggests the "signal" might just separate two extreme groups rather than measure a smooth underlying quantity, a materially less trustworthy story.

Worked example

Two signals both show decile 10 minus decile 1 average monthly returns of 1.2%. Signal A's ten decile averages, in order, are 0.1%, 0.2%, 0.35%, 0.5%, 0.6%, 0.75%, 0.9%, 1.0%, 1.1%, 1.3%, strictly increasing at every step. Signal B's are 0.1%, 0.3%, 0.9%, 0.4%, 0.6%, 0.5%, 0.8%, 0.7%, 1.0%, 1.3%, the same start and end point, and roughly the same average slope, but decile 3 exceeds decile 4, and decile 7 exceeds deciles 8 and 9 come back down before the final jump. Signal A passes a monotonicity check cleanly; Signal B would fail one, despite reporting an identical headline decile spread, a reader shown only the spread number would never know the two signals behave so differently in between.

What this means in practice

Monotonicity checks are a standard part of factor and signal due diligence precisely because the headline decile-spread statistic is easy to produce and easy to overstate: a wide spread built almost entirely from decile 10 or decile 1 alone, with everything in between flat or reversed, is a common and easily missed red flag in factor research. Reporting the full decile table (or its plot) alongside the top-minus-bottom spread, rather than the spread alone, is the simplest way to let a reader judge for themselves whether a signal behaves the way its headline number implies.